Their Greatest Hits (1971–1975) is the first compilation album by the Eagles, released in 1976. The album contains a selection of songs from the Eagles’ first four albums released in the period from the Eagles’ formation in 1971 up to 1975. It was the best-selling album of the 20th century in the United States, and it stayed the best-selling album in the U.S. for some years until it was surpassed by Michael Jackson’s Thriller after the artist’s death in 2009. It is the second highest-certified album by the Recording Industry Association of America at 29× Platinum, behind Thriller.

In 2017, it was selected for preservation in the National Recording Registry by the Library of Congress as being “culturally, historically, or artistically significant.”

Their Greatest Hits (1971–1975) comprises nine singles released between 1972 and 1975, plus the album track “Desperado.” All of these singles except “Tequila Sunrise” charted in the top 40, with five in the top ten, and “One of These Nights” and “Best of My Love” both topping the singles chart.

The manager of Eagles, Irving Azoff, said: “We decided it was time to put out the first greatest-hits because we had enough hits.” However, according to Don Felder, none of the band members had any say in the decision to release the compilation album. The band complained that the album was “nothing more than a ploy by the record company to sell product without having to pay additional production costs”. Don Henley was unhappy that songs like “Tequila Sunrise” and “Desperado” were lifted out of the context of original album in a way that he thought detrimental to the nature, quality and meaning of the music. He said: “All the record company was worried about were their quarterly reports. They didn’t give a shit whether the greatest hits album was good or not, they just wanted product.” Despite being unhappy with the album’s release, the band nevertheless reasoned that it gave them more time to work on the Hotel California album.